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Spectra of MHD Turbulenceand Particle Transportin Nonlinear Shocks
Andrey Vladimirov1 Andrei Bykov2 Don Ellison1
1 North Carolina State University
2 Ioffe Physical-Technical Institute
Nonlinear Processes in Astrophysical Plasmas5th Korean Astrophysics Workshop, APCTP
November 19, 2009
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 1 / 19
Introduction Goals
Goals
Construct a model of magnetic field amplification (MFA) in nonlinearshocks that describes in detail the spectra of magnetic fluctuations;
Focus on self-consistent modeling of the non-linear connectionsbetween the system components (thermal plasma, turbulence,superthermal particles);
Here, I will demonstrate a nonlinear shock model with B-field amplifiedsimultaneously by three different mechanisms (resonant andnon-resonant);
Two models of particle diffusion in strong magnetic turbulence will bepresented and compared.
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 2 / 19
Introduction Method
Method
The Nonlinear Model
Particle transport modeled with a Monte Carlo simulation;
Analytic, semi-phenomenological description for magnetic fieldamplification, self-consistently coupled to CR distribution and MHD flow;
Fundamental conservation laws used to iteratively derive a nonlinearshock modification that conserves mass, momentum and energy;
Reasoning
We describe a large dynamic range in turbulence scales and particleenergies;
Elements of the model tested against spacecraft observations ofheliospheric shocks;
Works for highly anisotropic particle distributions (particle escape andinjection; large gradients of u and B).
Ability to incorporate non-diffusive particle transport (future work).
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 3 / 19
Magnetic Field Amplification Overview of Plasma Instabilities Induced by Streaming Cosmic Rays
Plasma Instabilities Induced by CR Streaming
Bell’s nonresonantcurrent-driven instabilityworks on small scales bystretching magnetic fieldlines (Bell, 2004).
Resonant CR streaming in-stability. Alfven wavesgain energy from resonantparticles streaming fasterthan the waves. (Skilling,1975).
Nonresonant long-wavelength instabilityarises from the back-reaction of the amplifiedwaves on the current ofthe streaming CRs (Bykovet al., 2009).
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 4 / 19
Magnetic Field Amplification Our Analytical Model of Magnetic Field Amplification
Evolution of Waves in the Precursor
Definitions
We describe turbulence by W (x , k) – spectral energy density of turbulentfluctuations, and separate it into
W = WM + WK =X
i∈modes
W(i)M +
Xi∈modes
W(i)K .
WM – magnetic fluctuations, WK – associated plasma velocity fluctuations,and (i) runs over the three types of waves (A – Alfven waves, B – Bell’smodes, C – Bykov’s modes).
Equations
Evolution for each mode is given by the equation for W (i) = W(i)M + W
(i)K :
u∂W (i)
∂x= γ(i)W (i) − L(i) +
»−α(i)W (i) +
∂
∂k
“kW (i)
”–du
dx− ∂Π(i)
∂k(1)
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 5 / 19
Magnetic Field Amplification Resonant Streaming Instabilitiy
Resonant Streaming Instability
Quasi-Linear Theory
Growth rate at a wavenumber k is
γ(A)W (A) = vA
»∂Pcr(x , p)
∂x
˛dk
dp
˛–p=
eBlsck
;
Wavelengths resonant with CR particles: r−1g (pmax) < k < r−1
g (pmin);
Equal energy in magnetic and kinetic fluctuations.
Strong Fluctuations (Nonlinear Theory)
We include transit time damping (Lee & Volk 1973, Achterberg &Blandford, 1986):
L(A) =
rπ
2krg, th
hW (A)
i2
B20/(8π)
ωB .
Saturation at ∆B ≈ a few B0 may occur (Lucek & Bell 2000). We do notinclude this effect.
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 6 / 19
Magnetic Field Amplification Nonresonant Short Scale Current Driven Instablity (Bell)
Nonresonant Short Scale Current Driven Instability (Bell)
Quasi-Linear Theory
Growth rate at k is
γ(B) = vAk
r4πjdcB0k
− 1,
Amplifies short-scale fluctuations: k > r−1g (pmin);
Velocity fluctuations contain a few times more energy than magneticfluctuations.
Strong Fluctuations (Nonlinear Theory)
Simulationsa show for ∆B & B0:
Growth slows down (saturation at rg min ≈ k?):
Dominant k decreases (dissipation at large k, inverse cascade, or both)
We assume, following Riquelme & Spitkovsky 2009, nonlinear dissipation
L(B) = ChW (B)
i 32ρ−
12 k
32 .
aBell, Reville+, Zirakashvili+, Niemec+, Riquelme+
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 7 / 19
Magnetic Field Amplification Nonresonant Large Scale Current Driven Instablity (Bykov)
Nonresonant Large Scale Current Driven Instablity (Bykov)
Quasi-Linear Theory
Fastest growing mode has
γ(C) = vA
ra
η
4πjdcB0
k
Operates on large scales: k < r−1g (pmax);
Predominantly velocity fluctuations in the eigenmodes.
Strong Fluctuations (Nonlinear Theory)
Slow growth: does not enter the strongly nonlinear regime ∆B � B0;
Saturation via velocity fluctuations may be important.
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 8 / 19
Magnetic Field Amplification Relationship between the Instabilities
At any point in the precursor, the 3 instabilities operate in non-overlappingregions of k-space1.
10-2
10-1
1
10
(k),
yr-1
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
k/rg0-1
Growth Rate
ResonantNonresonant (Bell)Nonresonant (Bykov)
The short-wavelength instability (Bell) is the fastest, but short-scaleturbulence does not efficiently scatter particles;
The resonant instability is slower, but produces stronger stcattering;
The long-wavelength instability (Bykov) is slow, but amplifies waves thatmay be important for the highest energy (escaping?) particles.
1The small overlap visible in the plot is for continuity in the numerical scheme
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 9 / 19
Feedback of Waves on Particles Diffusion Coefficient
Calculating the Particle Diffusion Coefficient
Prescription
Our model for diffusion coefficient uses magnetic field re-normalization;
Reproduces known asympthotic behavior for some cases with ∆B � B0
and ∆B � B0, smoothly interpolates between them;
Details in AV’s dissertation.
Features
This prescription reproduces:
The λ ∝ p2 scattering in small-scale fluctuations for Bss � B0 (highestenergy particles)
The resonant scattering regime λ ∝ p2−s for W ∝ k−s (intermediateenergies)
Two Models
Two possibilities for diffusion of magnetized particles (rg < lcor):
Model A: λ = const for the smallest particle energies;
Model B: λ ∝ p for the smallest particle energies.
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 10 / 19
Feedback of Waves on Particles Diffusion Coefficient
Examples of Difusion Coefficient Calculation
Example 1.
Weak white noise W ∝ k−1 + uniform B0:
10-8
10-6
10-4
10-2
1
102
kW(k
)/(
B02 /8
)
10-5
10-3
10-1
10
k/rg0-1
Turbulence Spectrum
10
103
105
107
109
D(p
)/(c
r g0)
10 102
103
104
105
106
p/mpc
Diffusion Coefficient
BohmResonantOur models (A and B)
Our model reproduces the well known resonant scattering rate: ifW (k) ∝ k−α, then D(p) ∝ p2−α.
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 11 / 19
Feedback of Waves on Particles Diffusion Coefficient
Examples of Difusion Coefficient Calculation
Example 2.
Kolmogorov-like power law spectrum of turbulence W ∝ k−5/3:
10-8
10-6
10-4
10-2
1
102
kW(k
)/(
B02 /8
)
10-5
10-3
10-1
10
k/rg0-1
Turbulence Spectrum
10
102
103
104
105
106
107
108
109
D(p
)/(c
r g0)
10 102
103
104
105
106
p/mpc
Diffusion Coefficient
BohmResonantOur models (A and B)
Our model reproduces the well known resonant scattering rate: ifW (k) ∝ k−α, then D(p) ∝ p2−α.
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 11 / 19
Feedback of Waves on Particles Diffusion Coefficient
Examples of Difusion Coefficient Calculation
Example 3.
Strong, short scale turbulence:
10-8
10-6
10-4
10-2
1
102
kW(k
)/(
B02 /8
)
10-5
10-3
10-1
10
k/rg0-1
Turbulence Spectrum
10
103
105
107
109
D(p
)/(c
r g0)
10 102
103
104
105
106
p/mpc
Diffusion Coefficient
BohmResonantOur models (A and B)
Our model correctly describes the D(p) ∝ p2 behavior (notice how theresonant scattering and Bohm models fail).
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 11 / 19
Feedback of Waves on Particles Diffusion Coefficient
Examples of Difusion Coefficient Calculation
Example 4.
Synthetic spectrum:
10-2
10-1
1
10
102
kW(k
)/(
B02 /8
)
10-5
10-4
10-3
10-2
k/rg0-1
Turbulence Spectrum
10
103
105
107
109
D(p
)/(c
r g0)
102
103
104
105
106
p/mpc
Diffusion Coefficient
BohmResonantOur model AOur model BSimulation BR+ 2009
The spectrum and the orange data points are from the simulation by Reville etal. 2008. Our calculations agree in the range where D(p) increases with p.
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 11 / 19
Feedback of Waves on Particles Nonlinear Shock Simulation
Our model simulatesparticle acceleration,turbulence generationand shocked flow allconsistently with eachother;
A Monte Carlo (MC)code describes particletransport and acceleration;
Diffusion coefficient usedin the MC code coupled toturbulence spectrum;
Turbulence generationdriven according to particletransport simulated in MC.
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 12 / 19
Simulation Results Summary
Summary of a Set of Simulations
Model A. Magnetized particles have λ(p) = const
10-6
10-5
10-4
10-3
10-2
Bef
f,G
2000 5000 10000 20000
u0, km s-1
Amplified Magnetic Field
ResonantNonresonant (Bell)Nonresonant (Bykov)
4
5
6
7
8
9
10
11
Rto
t
2000 5000 10000 20000
u0, km s-1
Compression Ratio
107
108
109
1010
T2,
K
2000 5000 10000 20000
u0, km s-1
Downstream Temperature
SimulationRankine-Hugoniot
Physical parameters in these simulations represent a SNR shock expanding intoa cold interstellar medium (n0 = 0.3 cm−3, T0 = 104 K, B0 = 3 µG).Acceleration is size limited, DFEB = 0.03 pc, parallel geometry, steady state.
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 13 / 19
Simulation Results Summary
Summary of a Set of Simulations
Model B. Magnetized particles have λ(p) ∝ p
10-6
10-5
10-4
10-3
10-2
Bef
f,G
2000 5000 10000 20000
u0, km s-1
Amplified Magnetic Field
ResonantNonresonant (Bell)Nonresonant (Bykov)
4
5
6
7
8
9
10
11
Rto
t
2000 5000 10000 20000
u0, km s-1
Compression Ratio
107
108
109
1010
T2,
K
2000 5000 10000 20000
u0, km s-1
Downstream Temperature
SimulationRankine-Hugoniot
Physical parameters in these simulations represent a SNR shock expanding intoa cold interstellar medium (n0 = 0.3 cm−3, T0 = 104 K, B0 = 3 µG).Acceleration is size limited, DFEB = 0.03 pc, parallel geometry, steady state.
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 13 / 19
Simulation Results Turbulence Spectra
Energy spectra of magnetic fluctuations downstream of the u0 = 2500 km s−1
shock shown below.
Model A. Magnetized particles have λ(p) = const
10-1
1
10
102
103
kW(k
)/(
B02 /8
)
10-8
10-6
10-4
10-2
1 102
104
k/rg0-1
Turbulence Spectrum
AlfvenBellBykov
14
16
18
20
22
24
26
D(p
),cm
-2 -1 0 1 2 3 4
p/mpc
Diffusion Coefficient
Our ModelBohm
Growth rates of instabilities are finite in non-overlapping k-space regions, butan overlap in the spectra occurs due to the integration through the precursor.
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 14 / 19
Simulation Results Turbulence Spectra
Energy spectra of magnetic fluctuations downstream of the u0 = 2500 km s−1
shock shown below.
Model B. Magnetized particles have λ(p) ∝ p
10-1
1
10
102
103
kW(k
)/(
B02 /8
)
10-8
10-6
10-4
10-2
1 102
104
k/rg0-1
Turbulence Spectrum
AlfvenBellBykov
14
16
18
20
22
24
26
D(p
),cm
-2 -1 0 1 2 3 4
p/mpc
Diffusion Coefficient
Our ModelBohm
Growth rates of instabilities are finite in non-overlapping k-space regions, butan overlap in the spectra occurs due to the integration through the precursor.
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 14 / 19
Simulation Results Turbulence and Escaping Particles
Model A. Particle escape in the presence of all three instabilities
10-2
1
102
kW(k
)/(
B02 /8
)
10-8
10-6
10-4
10-2
1 102
104
k/rg0-1
Turbulence Spectrum
AlfvenBellBykov
-12.0
-11.5
-11.0
-10.5
-10.0
-9.5
-9.0
-8.5
-8.0
p4 f(p)
1.0 1.5 2.0 2.5 3.0 3.5 4.0
p/mpc
Escaping particle distribution function
Here, turbulence spectrum is shown at the upstream free escape boundary forthe shock with u0 = 2000 km s−1.Beff downstream in this case is dominated by Bell’s modes, but particle escapeupstream is shaped by the resonant and the long-wavelength modes.
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 15 / 19
Simulation Results Turbulence and Escaping Particles
Model A. Particle escape in the presence of only Bell’s instability
10-2
1
102
kW(k
)/(
B02 /8
)
10-8
10-6
10-4
10-2
1 102
104
k/rg0-1
Turbulence Spectrum
AlfvenBellBykov
-12.0
-11.5
-11.0
-10.5
-10.0
-9.5
-9.0
-8.5
-8.0
p4 f(p)
1.0 1.5 2.0 2.5 3.0 3.5 4.0
p/mpc
Escaping particle distribution function
Here, turbulence spectrum is shown at the upstream free escape boundary forthe shock with u0 = 2000 km s−1.Beff downstream in this case is dominated by Bell’s modes, but particle escapeupstream is shaped by the resonant and the long-wavelength modes.
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 15 / 19
Simulation Results Observational Features
Spectrum of Magnetic Fluctuations and Synchrotron Radiation
Figure: AV+, 2009, ApJL
⇔
Figure: Bykov+, 2009, MNRAS
In strong turbulence, fluctuating B changes direction as well as magnitude.This modifies emitted synchrotron spectrum and may lead to time variableX-ray dots and clumps (Bykov, Uvarov, & Ellison. 2008, MNRAS), dependingon the turbulence spectrum.
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 16 / 19
Discussion Summary of our Findings
Summary of our Findings
Model of Nonlinear DSA with Magnetic Field Amplification
Self-consistently couples particle acceleration, MFA and nonlinear flowmodification;
Combines 3 plasma instabilities (resonant and 2 non-resonant) inducedby accelerated particles;
Simulates particle transport based on the turbulence energy spectra.
Results
Macroscopic shock parameters (T2, Rtot, Beff , etc.) depend on theconnection between turbulence spectra and particle transport;
For high u0, Bell’s nonresonant instability dominates Beff and, therefore,determines the synchrotron emission;
Highest energy particles (esp. escaping upstream) are sensitive to theturbulence produced by the resonant and long-wavelength nonresonantmechanism;
Synchrotron emission modified by time-variability of B may be adiagnostic of turbulence spectra produced by SNR shocks.
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 17 / 19
Discussion Open Questions
Open Questions
How do the instabilities evolve in the nonlinear regime(∆B � B0)?
Nonlinear growth rate;
Wave dissipation;
Spectral energy transfer;
Other MFA mechanisms [e.g., dynamo (Beresnyak, Cho, Ryu...) ormagnetosonic instability (Malkov & Diamond), and other].
What happens at the subshock?
Transmission of turbulence;
Effect on particle injection.
Which model best describes low energy particle transport?
Is either of our models (λ ∝ p or λ = const) valid for rg < lcor?
Are time variability of B and velocity fluctuations important here?
May the particle transport be non-diffusive?
Non-stationary solutions?
E.g., work by Jones, Kang & Ryu and others.A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 18 / 19
Discussion References
Some References Mentioned in This Talk
Nonresonant short-wavelength instability: Bell, MNRAS 2004;
Nonresonant long-wavelength instability: Bykov, Osipov, & Toptygin,AstL 2009;
Dissipation of Bell’s instability: Riquelme & Spitkovsky, ApJ 2009;
Model of diffusion discussed here and the Monte Carlo method:Vladimirov, arXiv:0904.3760.v1 2009;
Particle transport simulation: Reville et al. MNRAS 2008;
Dissipation of Alfven waves: Lee & Volk, Ap&SS 1973; Achterberg &Blandford, MNRAS 1986
Nonlinear shocks with Bell’s instability: Vladimirov, Bykov, & Ellison,ApJL 2009;
Synchrotron emission in turbulent B-field: Bykov, Uvarov, & Ellison, ApJ2008; Bykov et al., MNRAS 2009, v. 399.
A. Vladimirov (NCSU) Turbulence in Nonlinear Shocks November 19, 2009 19 / 19